Venn Diagrams
Patch Kessler
2025.02.24
A Venn diagram gives a clear picture of all the ways of making a set of binary choices. In the following diagram, points on the plane are either inside or outside of three different circular regions.
There’s math in these diagrams, and also compelling graphic design. A beautiful survey of different forms of these diagrams was created recently by Hugh Dubberly, and the history of these diagrams is reviewed in some detail in Cogwheels of the Mind, by A.W.F. Edwards.
If there is just one choice, for instance between \(A\) and not \(A\) (which we denote by \(!A\)), we create a Venn diagram by using a loop to divide the plane into two regions, one for \(A\) and one for \(!A\).
To accommodate an additional choice, we draw a new loop that splits each of the existing regions (i.e., \(A\) and \(!A\)) in two.
Generally, to accommodate an additional choice, we draw a new loop that splits all the previous regions in two. The problem with doing this in the plane is that the loops necessarily have funny convoluted shapes. Here is a Venn diagram with four regions.
We would prefer the loops to be simple and consistent- for instance it would be great if they were all circles.
Another difficulty with working in the plane is that there is a big visual difference between the inside and the outside of a circle. Being inside a circle feels “correct”, while being outside the circle feels like something isn't quite right. Like you missed a target. However the choice represented by being inside and outside a circle might not have these associations.
Context Shift
I woke up Sunday morning thinking about Venn diagrams, and about how there must be a way to build them in general with circles. Or perhaps with spheres. As an example, here's a four region Venn diagram consisting of four intersecting spheres in 3D.
Not a great drawing, but the idea is that spheres with unit radius have their centers at the vertices of a tetrahedron all edges of which have unit length. After spending some time thinking about how to generalize this to higher dimensions, I realized things get much simpler if the working space is curved rather than flat. For instance, with three regions, instead of working on a 2D plane, we can work on the 2D surface of a sphere.
Circles in the plane become great circles on the sphere. The three great circles on this sphere divide it into eight regions that are exactly the same. Compare this to working in the plane, where the regions created by the intersecting circles are dramatically different.
The above Venn diagram is for 3 binary choices. In general, with \(n\) binary choices, we work on the surface of the hypersphere \(S^{n-1}\subset\mathbb{R}^n\), which consists of all the points in \(\mathbb{R}^n\) that are a distance 1 from the origin.
$$
S^{n-1} = \{x\in\mathbb{R}^n: x^Tx = 1\}
$$
Where \(x_i\) is the \(i^{th}\) component of \(x\), the hyperplane \(x_i=0\) intersects \(S^{n-1}\) to give a (hyper) great circle. This great circle divides \(S^{n-1}\) into two regions, corresponding to the \(i^{th}\) binary choice.
In general, the mental image we have for Venn diagrams is of a (hyper) sphere divided into identical regions by (hyper) great circles. It's refreshingly clean and simple!
